The Einstein Relation on Metric Measure Spaces

This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at H\"older regular transformations and how they influence the local walk dimension and prove some partial results concerning the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.Comment: 28 pages, 3 figure

Minkowski Content and local Minkowski Content for a class of self-conformal sets

We investigate (local) Minkowski measurability of $\mathcal C^{1+\alpha}$ images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set $K$ implies (local) Minkowski measurability of its image $F$ and provide an explicit formula for the (local) Minkowski content of $F$ in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, $F$ can be Minkowski measurable although $K$ is not. However, we obtain that an average version of the (local) Minkowski content of both $K$ and $F$ always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of $K$ and $F$.Comment: The final publication is available at http://www.springerlink.co